Determining formation gas composition during well drilling

ABSTRACT

Some aspects determining formation gas composition during well drilling can be implemented as a computer-implemented method, a computer-readable medium, or a computer system. A theoretical diffusion coefficient for a drilling fluid that comprises gas from a formation through which the drilling fluid is flowed is determined. The theoretical diffusion coefficient is based on an extraction of all of the gas from the drilling fluid. An experimental diffusion coefficient for the drilling fluid based on well drilling parameters is determined. A concentration of the gas at the formation is determined based, at least in part, on a difference between the theoretical diffusion coefficient and the experimental diffusion coefficient. The determined concentration at the formation is provided.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a U.S. National Phase Application under 35 U.S.C. §371 and claims the benefit of priority to International Application Serial No. PCT/US2013/069718, filed on Nov. 12, 2013, the contents of which are hereby incorporated by reference.

TECHNICAL FIELD

This disclosure relates to fingerprinting formation gas in a hydrocarbon reservoir.

BACKGROUND

Hydrocarbon fingerprinting is a process by which hydrocarbons (oil or gas or both) are defined into its components in such a way as to permit the identification of a particular sample of the hydrocarbons by the uniqueness of its composition. Fingerprinting can be implemented to identify source reservoirs from which the hydrocarbons are taken. For example, when a new well intersects hydrocarbons in a reservoir, fingerprinting can be implemented to determine if the new well lies in a new reservoir or in an extension of a previously discovered reservoir (e.g., in an offset well or in a series of offset wells). Doing so can enable mapping the extent of a reservoir and estimating the reservoir's size. Fingerprinting can also be used to facilitate the commingling of hydrocarbons from more than one reservoir through a common well. Fingerprinting allows the hydrocarbons from multiple wells to be commingled and the respective contributions to be identified by source and proportion. Such fingerprinting can be implemented when drilling a well in a formation, e.g., by evaluating the hydrocarbons in the formation as the well is being drilled.

DESCRIPTION OF DRAWINGS

FIG. 1 illustrates an example well being drilled.

FIG. 2 is a flowchart of an example process for determining formation gas concentration in the example well of FIG. 1.

FIG. 3 illustrates an example architecture of an example computer system that implements the example process of FIG. 2.

Like reference symbols in the various drawings indicate like elements.

DETAILED DESCRIPTION

This disclosure relates to determining formation gas composition in a subterranean zone while drilling a well. Formation gas (i.e., gas trapped in the formation), which is released when the well is being drilled, can be composed of different gases (e.g., hydrocarbons) at different concentrations. During drilling, at least some of the formation gas is carried by the drilling fluid from the formation outside of the well. The formation gas can be extracted from the drilling fluid and then evaluated, e.g., to determine the constituent species and respective concentrations in the formation gas. A gas extraction device can be implemented to extract the formation gas from the drilling fluid.

This disclosure describes implementing a mathematical model based on Fick's laws of diffusion that can correct for deficiencies in formation gas extraction from the drilling fluid. In some implementations, either Fick's First Law or Fick's Second Law (or both) is solved for a first, theoretical diffusion constant of the formation gas using a theoretical perfect mass flux that is based on 100% extraction of the formation gas from the drilling fluid. A second, experimental diffusion constant is determined using an experimental mass flux from the gas extraction device. The difference between the two values is then used to correct the experimental data to the experimental mass flux. In this manner, the diffusion constants are determined as a function of the chemical or physical properties (or both) of the drilling fluid.

Knowing gas extraction efficiency can enable analyzing the extracted formation gas to determine the composition and concentration of each species in the formation gas. Knowing the gas extraction efficiency can enable correcting gas data to reflect the true chemical composition of formation fluid. Knowing such composition and concentration at different locations (e.g., depths) in the well can enable fingerprinting the formation gas. Such fingerprinting can be used to determine, e.g., if the formation gas is in a new gas reservoir or is part of an existing gas reservoir (e.g., in an offset well or a series of offset wells). Further, knowing the composition and concentration of the formation gas at different positions in the well can also enable characterizing the formation through which the well is being drilled. Moreover, understanding the variation of diffusion constants with different drilling fluids can allow for extrapolation and interpolation between physical and chemical properties of different drilling fluids with minimal experimentation.

FIG. 1 illustrates an example well being drilled. In some implementations, a drill string 104 (e.g., a cylindrical drill string) can be disposed in a well 102 being drilled through a subterranean zone of interest. A subterranean zone of interest can include a single formation, portions of a single formation, multiple formations (or combinations of them). A well drilling tool (e.g., a drill bit) can be attached to a downhole end of the drill string 104 and operated to drill the well 102. Drilling fluid can be flowed from the surface into the drill string 104 through a wellhead 106 for many purposes including, e.g., lubricating the well drilling tool, carrying formation cuttings and other debris from the well drilling tool to the surface, and other purposes. The drilling fluid can flow to the well drilling tool through the drill string 104, into an annulus defined between the drill string 104 and an inner wall of the well 102, and to the surface through the wellhead 106. As the well drilling tool is operated to drill the well 102, formation gas can be released from the portion of the formation being cut by the well drilling tool. The well drilling fluid can be flowed with sufficient pressure to carry at least a portion of the formation gas released during drilling out of the well 102.

A gas extraction device 108, positioned at the surface to receive the well drilling fluid that carries the formation gas, can extract the formation gas from the well drilling fluid. The extracted formation gas can be provided to a gas analyzer 110 which can, e.g., identify the components in the formation gas and the respective concentration of each component. In some implementations, the gas analyzer 110 can be a gas chromatography mass spectrometer (GCMS), a liquid chromatography mass spectrometer (LCMS), a combination of the two, and/or a device that implements other analytical techniques to analyze the formation gas. The output of the gas analyzer 110 (e.g., the components in the formation gas and the concentrations of the components) are provided to a computer system 112 that implements techniques to determine concentrations of the formation gas at the downhole location (i.e., near the well drilling tool) as described below with reference to FIG. 2.

FIG. 2 is a flowchart of an example process 200 for determining formation gas concentration in the example well of FIG. 1. In some implementations, at least a portion of the process 200 can be implemented by the computer system 112 as computer instructions stored on a computer-readable medium 114 and executable by one or more processors 116.

At 202, sample data describing the drilling fluid carrying the formation gas can be obtained. In some implementations, the sample data can be obtained at or near a downhole location (e.g., at which the well drilling tool is drilling), while, in others, the sample data can be obtained at a surface of the well 102 (e.g., before the drilling fluid is flowed to the gas extraction device 108). Also, as described below, the sample data can be obtained over a period of time and the sampling data, which includes, e.g., recombined reservoir production data or downhole reservoir data, or both) can be normalized to represent formation gas from a specified volume (e.g., a cubic foot) of formation.

In implementations in which the sample data is obtained at or near the downhole location, and in implementations in which the sample data is obtained at the surface, the computer system 112 can determine the theoretical diffusion coefficient as described below. The computer system 112 can receive inputs including a drilling fluid flow rate into and out of the well 102, a rate of penetration (ROP) of the well drilling tool, volumes of cuttings, an estimated porosity of the formation, and other inputs. From the inputs, the computer system 112 can determine the flux for the gas extraction device 108 using liquid surface area, formation gas flow rate, drilling fluid flow rate, and average drilling fluid thickness for theoretical complete extraction.

At 204, the computer system 112 can determine a theoretical diffusion coefficient for a drilling fluid that comprises a gas from a formation through which the drilling fluid is flowed based on either Fick's First law of diffusion (Equation 1) or Fick's Second law of diffusion (Equation 2).

$\begin{matrix} {J = {{- \rho}\; D\frac{\partial\phi}{\partial t}}} & \left( {{Equation}\mspace{14mu} 1} \right) \\ {\frac{\partial\phi}{\partial t} = {D\frac{\partial^{2}\phi}{\partial t^{2}}}} & \left( {{Equation}\mspace{14mu} 2} \right) \end{matrix}$

To solve Fick's laws of diffusion, the computer system 112 can receive, as input, flux (J), drilling fluid density (ρ), an assumed or measured distance of travel for the fluid, an area, and change in concentration over change in time (∂φ/∂t). The computer system 112 can be implemented to solve Fick's laws of diffusion as needed for the dimensions and times of interest for the theoretical diffusion constants for each gas species in the formation gas. The computer system 112 can determine a theoretical diffusion coefficient as described below.

Fick's 1^(st) law of diffusion is represented by Equation 1a.

$\begin{matrix} {\frac{\omega_{Ay}}{A} = {{- \rho}\; D_{AB}\frac{\omega_{A\; 0} - 0}{y}}} & \left( {{Equation}\mspace{14mu} 1a} \right) \end{matrix}$

In Equation 1a, ω_(Ay) represents mass flow rate, A represents area, ρ represents bulk density, D_(AB) represents diffusion of species A through substance B, and ω_(A0) represents mass fraction of species A at time, t=0 (i.e., initial concentration of species A). Equation la represents a modification of three-dimensional Fick's law (Equation 1) by assuming no flux in the Z direction. Equation la is represented in the Cartesian coordinates system, but can alternatively be represented in spherical or cylindrical coordinates.

Molecular mass flux, j_(Ay), is represented by ω_(Ay)/A. The change in concentration over time, dω_(A)/dy, is represented as shown in Equation 1b.

$\begin{matrix} {\frac{\partial\omega_{A}}{\partial y} = \frac{\omega_{A\; 0} - 0}{y}} & \left( {{Equation}\mspace{14mu} 1b} \right) \end{matrix}$

In Equations 1a and 1b, density is the density of fluid (e.g., water or oil-based drilling fluid). Species B is treated as the bulk fluid and not each individual chemical species. It is assumed that the chemical species is instantaneously removed from the system once the chemical species leaves the bulk fluid into the gas phase. Laminar or turbulent flow is ignored in the liquid or gas phase. The surface area being a vortex, solving directly for the surface area may not be possible. Consequently, the area is approximated as a surface area of a cylinder and a cone combined (Equation 1c).

Area=2πr ²+2πrh ₁+(πr ² h ₂)/3  (Equation 1c)

In Equation 1c, r represents the radius of the vessel (e.g., drill string, or other tubing), h₁ represents the height of the cylinder and h₂ represents the height of the cone. The heights of the cylinder and the cone can be physically measured.

The mass flow rate, ω_(Ay), is calculated based on gas pump rates. As a first step, each species component is measured analytically. Mass concentration in the extraction vessel can then be calculated using Equation 1d.

$\begin{matrix} {\frac{P_{1}V_{1}}{T_{1}} = \frac{P_{2}V_{2}}{T_{2}}} & \left( {{Equation}\mspace{14mu} 1d} \right) \end{matrix}$

In Equation 1d, P₁, V₁, and T₁ represent pressure, volume, and temperature, respectively.

For each component species, mass concentration can be determined using the ideal gas laws represented by Equations 1e, 1f and 1g.

$\begin{matrix} {{PV} = {ZnRT}} & \left( {{Equation}\mspace{14mu} 1e} \right) \\ {n = {\frac{m}{M} = \frac{ZPV}{RT}}} & \left( {{Equation}\mspace{14mu} 1f} \right) \\ {m = {M\frac{ZPV}{RT}}} & \left( {{Equation}\mspace{14mu} 1g} \right) \end{matrix}$

Mass concentrations in the gas phase from extractor to detector can be determined based on Conservation of Mass laws. As an alternative to implementing Equations 1d-1g, the mass flow rate can be determined by numerical integration of the gas curve.

The mass flow rate, ω_(Ay) can be divided by area to obtain molecular mass flux, j_(Ay). At 206, the computer system 112 can determine an experimental diffusion coefficient of the formation gas. In implementations in which the sample data is obtained at or near the downhole location, the computer system 112 can determine the experimental diffusion coefficient as described below. The computer system 112 can determine a mass flux for Fick's first law of diffusion as shown in Equation 3a or 3b.

$\begin{matrix} {J_{\overset{.}{A}y} = \frac{\,^{\cdot}{mass}}{{area} \times {time}}} & \left( {{Equation}\mspace{14mu} 3a} \right) \\ {J_{\overset{.}{A}y} = \frac{\,^{.}{moles}}{{area} \times {time}}} & \left( {{Equation}\mspace{14mu} 3b} \right) \end{matrix}$

The computer system 112 can implement operations to define a specified volume (e.g., 1 liter or other volume), convert the specified volume to mass using density. Alternatively, or in addition, the computer system 112 can convert the specified volume to moles by first converting the volume to mass using density, and then converting the mass to moles using Avogadro's number (6.0221×10²³ mol⁻¹. The computer system 112 can implement operations to evaluate ideal gas laws or modified PVT (i.e., pressure, volume, temperature) relationships to determine mass of the formation gas at the gas extraction device 108 in gas phase. The computer system 112 can further implement an experimental model to determine surface area of fluid in the gas extraction device 108 by using an estimated shape. The computer system 112 can determine a resonance time for fluid in the gas extraction device 108. From the afore-described information, the computer system 112 can determine the mass flux.

Drilling fluid density (ρ) obtained by any density measurement method can be provided as an input to the computer system 112. The average thickness of the fluid is represented by Equation 4.

∂y=y _(i) −y ₀  (Equation 4)

The computer system 112 can determine the average thickness using the model or shape that was described above as being used to determine the surface area of the fluid in the gas extraction device 108.

The computer system 112 can determine the change in concentration (∂φ) by implementing Equation 5.

∂φ=φ^(measured)−φ^(initial)  (Equation 5)

In gas phase, φ^(measured) is zero, and, assuming perfect diffusion, is the same at the gas-liquid interface and in the fluid. It can be assumed that the formation gas is moving so fast as to refresh the surface to zero concentration. The computer system 112 can solve equations 3a, 3b, 4 and 5 to determine an experimental diffusion coefficient (D).

In implementations in which the sample data is obtained at the surface of the well 102, the computer system 112 can determine the experimental diffusion coefficient by implementing operations similar to those described above. The total formation gas in the drilling fluid can be determined by passing a sample of the drilling fluid through the gas extraction device 108, and by passing the gas extracted from the sample through the gas analyzer 110 that implements a GCMS, a LCMS, a combination of the two or other analytical technique.

Having determined the experimental diffusion coefficient, the computer system 112 can implement operations to determine what the mass flux should have been when the formation gas was released from the formation. Based on an estimated or measured porosity, φ, the computer system 112 can determine mass of hydrocarbons per unit of formation. It can be assumed that all pore spaces have been compromised and that there is little or no contribution from any formation besides the formation being drilled. Using the drilling fluid flow rate and the ROP, the computer system 112 can determine a concentration per mass of fluid using the composition of the formation gas obtained, e.g., from the gas analyzer 110. The computer system 112 can further determine mass per time (resonance time) in the gas extraction device 108 assuming that all mass moves from fluid to gas phase. Using the surface area determined as described above, the computer system 112 can determine a new flux, which represents the non-experimental flux or theoretical flux that would occur in an ideal extraction. Using the density of drilling fluid, the average thickness of the drilling fluid, and the change in concentration described above, and/or using external data (e.g., data obtained from the gas analyzer 110), the computer system 112 can determine a diffusion coefficient.

The diffusion coefficient, D_(AB), can be determined using Equation 6.

$\begin{matrix} {D_{AB} = {\frac{j_{Ay}}{\rho \frac{\omega_{A\; 0}}{\partial y}} = \frac{j_{Ay}{\partial y}}{A\; \rho \; \omega_{A\; 0}}}} & \left( {{Equation}\mspace{14mu} 6} \right) \end{matrix}$

The computer system 112 can implement the techniques described above to determine a diffusion coefficient for each species of interest. The diffusion coefficient is a function of fluid properties. In some implementations, testing can be done on a change in viscosity basis or a composition of base fluid, or both.

In some implementations, the computer system 112 can determine the theoretical and experimental diffusion coefficients using Fick's Second law of diffusion (Equation 2). The computer system 112 can determine the diffusion coefficient by implementing operations similar to those described above with reference to the new flux. Assuming drilling fluid circulation in a closed well, the computer system 112 can determine a change in concentration over a change in time (∂φ/ηt) by solving Equation 7.

$\begin{matrix} {\frac{\partial\phi}{\partial t} = \frac{\phi^{i} - \phi^{0}}{t^{i} - t^{0}}} & \left( {{Equation}\mspace{14mu} 7} \right) \end{matrix}$

Equation 6 can be solved under the boundary conditions represented by Equations 8a and 8b.

At Y=∞; φ=0  (Equation 8a)

At y=L (thickness); φ=measured value  (Equation 8b)

The computer system 112 can implement an integration operation on Equation 6 under the boundary conditions specified in Equations 7a and 7b to obtain the diffusion coefficient.

Returning to FIG. 2, at 208, the computer system 112 can determine a concentration of the formation gas based, at least in part, on a difference between the theoretical diffusion coefficient and the experimental diffusion coefficient. The differences in the diffusion constants represent an inefficiency of extraction by the gas extraction device 108, and the dependence of the diffusion constants on physical and chemical properties of the drilling fluid. In some implementations, the computer system 112 can determine a ratio of the experimental diffusion coefficient and the theoretical diffusion coefficient, and implement real-time drilling adjustment based on the ratio, as described below.

At 210, the computer system 112 can provide the determined concentration of the formation gas, e.g., as an output to an output device (e.g., a monitor, a printer, other output device) connected to the computer system 112 or as an input to a computer software application implemented by the computer system 112 to determine the diffusion constants in real-time, as described below.

In some implementations, the computer system 112 can identify physical properties of the drilling fluid other than density as well as chemical properties of the drilling fluid over a period of time at multiple instants separated by specified time intervals. At each time instant, the computer system 112 can determine the experimental and theoretical diffusion coefficients by implementing operations such as those described above. The computer system 112 can implement real-time operations to determine changes in drilling fluid properties to constantly adjust the diffusion coefficients for the formation gas. For example, at each instant, the computer system 112 can determine the ratio of the experimental and theoretical diffusion coefficients described above. In real-time, the computer system 112 can determine changes to the drilling fluid properties (e.g., drilling fluid flow rate, drilling fluid density, or other properties) or the drilling parameters (e.g., weight on bit, rate of penetration, or other parameters) that can result in the experimental and theoretical diffusion coefficients being as close to each other as possible.

Additionally, the computer system 112 can determine the concentration of the formation gas at each instant based, at least in part, on a difference between the experimental and theoretical diffusion coefficients determined for the instant. The concentration of the formation gas at each instant and a location of the well drilling tool at each instant can enable characterizing the formation.

FIG. 3 illustrates an example architecture of an example computer system that implements the example process of FIG. 2. The example computer system 112 can be located at or near one or more wells and/or at a remote location. The example computer system 112 includes the one or more processors 116, a computer-readable medium 114 (e.g., a memory), and input/output controllers 302 communicably coupled by a bus 165. The computer-readable medium can include, for example, a random access memory (RAM), a storage device (e.g., a writable read-only memory (ROM) and/or others), a hard disk, and/or another type of storage medium. The computer system 112 can be preprogrammed and/or it can be programmed (and reprogrammed) by loading a program from another source (e.g., from a CD-ROM, from another computer device through a data network, and/or in another manner). The input/output controller 170 is coupled to input/output devices and to a network 304. The input/output devices receive and transmit data in analog or digital form over communication links such as a serial link, wireless link (e.g., infrared, radio frequency, and/or others), parallel link, and/or another type of link.

The network 172 can include any type of data communication network. For example, the network 172 can include a wireless and/or a wired network, a Local Area Network (LAN), a Wide Area Network (WAN), a private network, a public network (such as the Internet), a WiFi network, a network that includes a satellite link, and/or another type of data communication network.

A number of implementations have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the disclosure. 

1. A computer-implemented method comprising: determining a theoretical diffusion coefficient for a drilling fluid comprising a gas from a formation, the theoretical diffusion coefficient determined based on an extraction of all of the gas from the drilling fluid; determining an experimental diffusion coefficient for the drilling fluid based on well drilling parameters including a flow rate of the drilling fluid through the well; determining a concentration of the gas at the formation based, at least in part, on a difference between the theoretical diffusion coefficient and the experimental diffusion coefficient; and providing the determined concentration of the gas at the formation.
 2. The method of claim 1, wherein determining the theoretical diffusion coefficient comprises solving Fick's first law of diffusion or Fick's second law of diffusion for the theoretical diffusion coefficient.
 3. The method of claim 1, wherein determining the theoretical diffusion coefficient and the experimental diffusion coefficient comprises determining the theoretical diffusion coefficient and the experimental diffusion coefficient from sample data describing the drilling fluid carrying the gas near the formation.
 4. The method of claim 1, wherein determining the theoretical diffusion coefficient and the experimental diffusion coefficient comprises determining the theoretical diffusion coefficient and the experimental diffusion coefficient from sample data describing the drilling fluid carrying the gas at the surface.
 5. The method of claim 4, wherein determining the theoretical diffusion coefficient and the experimental diffusion coefficient from sample data comprises measuring the concentration of the gas in a sample of the drilling fluid using one or more of gas chromatography, liquid chromatography, and mass spectrometry.
 6. The method of claim 1, wherein determining the experimental diffusion coefficient comprises: identifying physical and chemical properties of the drilling fluid at an instant in time; and determining the experimental diffusion coefficient for the drilling fluid having the identified physical and chemical properties at the instant in time.
 7. The method of claim 6, wherein determining the experimental diffusion coefficient comprises providing the identified physical and chemical properties at the instant in time to a mathematical model that determines the experimental coefficient based on the identified physical and chemical properties at the instant in time.
 8. The method of claim 1, further comprising determining a plurality of experimental diffusion coefficients of the gas at the formation at a respective plurality of sequential instances in time.
 9. The method of claim 8, further comprising periodically determining the concentration of the gas at the formation based, at least in part, on a plurality of differences between the theoretical diffusion coefficient and the plurality of experimental diffusion coefficients.
 10. The method of claim 1, further comprising: determining a plurality of concentrations of the gas at a corresponding plurality of formations, each concentration of gas determined based, at least in part, on a difference between the theoretical diffusion coefficient and a corresponding experimental diffusion coefficient determined for each formation; comparing the plurality of concentrations with each other, determining that a first concentration at a first formation is similar to a second concentration at a second formation; and determining that the first formation is similar to the second formation based on determining that the first concentration at the first formation is similar to the second concentration at the second formation.
 11. A non-transitory computer-readable medium storing instructions executable by one or more processors to perform operations comprising: determining a theoretical diffusion coefficient for a drilling fluid comprising a gas from a formation, the theoretical diffusion coefficient determined based on an extraction of all of the gas from the drilling fluid; determining an experimental diffusion coefficient for the drilling fluid based on well drilling parameters including a flow rate of the drilling fluid through the well; determining a concentration of the gas at the formation based, at least in part, on a difference between the theoretical diffusion coefficient and the experimental diffusion coefficient; and providing the determined concentration of the gas at the formation.
 12. The medium of claim 11, wherein determining the theoretical diffusion coefficient comprises solving Fick's first law of diffusion or Fick's second law of diffusion for the theoretical diffusion coefficient.
 13. The medium of claim 11, wherein determining the theoretical diffusion coefficient and the experimental diffusion coefficient comprises determining the theoretical diffusion coefficient and the experimental diffusion coefficient from sample data describing the drilling fluid carrying the gas near the formation.
 14. The medium of claim 11, wherein determining the theoretical diffusion coefficient and the experimental diffusion coefficient comprises determining the theoretical diffusion coefficient and the experimental diffusion coefficient from sample data describing the drilling fluid carrying the gas at the surface.
 15. The medium of claim 14, wherein the sample data comprises the concentration of the gasmeasured using one or more of gas chromatography, liquid chromatography, or mass spectrometry.
 16. A system comprising: one or more processors; and a computer-readable medium storing instructions executable by the one or more processors to perform operations comprising: determining a theoretical diffusion coefficient for a drilling fluid comprising a gas from a formation, the theoretical diffusion coefficient determined based on an extraction of all of the gas from the drilling fluid; determining an experimental diffusion coefficient for the drilling fluid based on well drilling parameters including a flow rate of the drilling fluid through the well; determining a concentration of the gas at the formation based, at least in part, on a difference between the theoretical diffusion coefficient and the experimental diffusion coefficient; and providing the determined concentration of the gas at the formation.
 17. The system of claim 16, wherein determining the experimental diffusion coefficient comprises: identifying physical and chemical properties of the drilling fluid at an instant in time; and determining the experimental diffusion coefficient for the drilling fluid having the identified physical and chemical properties at the instant in time.
 18. The system of claim 17, wherein determining the experimental diffusion coefficient comprises providing the identified physical and chemical properties at the instant in time to a mathematical model that determines the experimental coefficient based on the identified physical and chemical properties at the instant in time.
 19. The system of claim 16, the operations further comprising determining a plurality of experimental diffusion coefficients of the gas at the formation at a respective plurality of sequential instances in time.
 20. The system of claim 19, the operations further comprising periodically determining the concentration of the gas at the formation based, at least in part, on a plurality of differences between the theoretical diffusion coefficient and the plurality of experimental diffusion coefficients. 